5757 S. University Ave.

Chicago, IL 60637

Main: 773.702.5599

bfi.uchicago.edu

WORKING PAPER · NO. 2020-94

Technology DiffusionNancy StokeyJULY 2020

Technology DiffusionNancy StokeyJuly 2020JEL No. O14,O33

ABSTRACT

The importance of new technologies derives from the fact that they spread across many different users and uses, as well as different geographic regions. The diffusion of technological improvements, across producers within a country and across international borders, is critical for long run growth. This paper looks at some evidence on adoption patterns in the U.S. for specific innovations, reviews some evidence on the diffusion of new technologies across international boundaries, and looks at two theoretical frameworks for studying the two types of evidence. One focuses on the dynamics of adoption costs, the other on input costs.

Nancy StokeyDepartment of EconomicsUniversity of Chicago1126 East 59th StreetChicago, IL 60637and NBERnstokey@uchicago.edu

1. INTRODUCTION

Sustained long-run growth requires the adoption of new technologies.1 Thus, in-

novation, whether it is costly R&D or serendipitous discovery, is fundamental for

growth and has|deservedly|been well studied. But the importance of most new

technologies derives from the fact that they spread across many di�erent users and

uses, as well as di�erent geographic regions. Thus, the di�usion of technological im-

provements, across producers within a country and across international borders, is

arguably as critical as innovation for long run growth. Technology di�usion is the

focus here.

Good data on di�usion are not readily available. Indeed, for many innovations,

there are none at all. This paper looks at the evidence on adoption patterns and

rates in the U.S. for several speci�c innovations where good micro data have permit-

ted detailed studies. It then reviews some of the evidence on the di�usion of new

technologies across international boundaries, where data is even more limited. No

attempt is made to review all the work on technology adoption.

The discussion is selective and focuses on the role of cost reduction. Speci�cally,

two aspects of cost are considered. The �rst involves the dynamics induced by changes

in the �xed cost of adoption. Adoption takes time, and economic motives govern who

adopts a new technology and how quickly they adopt it. As use of an innovation

increases, its quality typically improves and its cost of adoption falls. Consequently,

early adoption by some users facilitates later adoption by a broader set of users. The

dynamics of adoption costs are important for explaining di�usion across users in a

1The Solow residual, the part of output growth that is unexplained by measurable inputs, is very

large for all developed countries. The same is true for share of output di�erences across countries.

Like any residual, Solow residual picks up the e�ect of anything that is omitted. But a substantial

component is surely technological change.

2

single environment.

The second aspect of cost involves relative input prices. Many new technologies

are, by design, labor-saving and capital-using, so their attractiveness depends on

the relative prices of capital and labor inputs. Wage rates vary enormously across

countries, while the cost of capital varies much less. Hence relative input prices are

important for explaining di�usion across countries.

Direct evidence on adoption patterns across countries is rarely collected, but indi-

rect evidence is sometimes available. Many technologies are `embodied' in new capital

goods, speci�c to them. This fact is useful, since good data are available on invest-

ments in tangible capital. Moreover, technologies that are embodied in capital goods

have a unique method for international di�usion: the capital goods themselves are

highly traded. Thus, equipment imports are a channel by which one country|either

developed or developing|can acquire technology from abroad. Producer equipment

is highly traded, and for developing countries a large fraction of their total investment

in producer equipment consists of imported goods coming from advanced countries.

Moreover, there are good data on the source of the equipment, as well as type, by

fairly narrowly de�ned sector.2

Before proceeding, two limits on the scope of this study should be noted. First, only

producer technologies are considered here. New consumer goods are also important

for welfare, but their di�usion is explained by a di�erent set of factors. For the same

reason, there will be no discussion of adoption of high-yield varieties (HYV's) in India,

sub-Saharan Africa and elsewhere.

Finally, note that the focus here is on di�usion of technologies, as opposed to ideas.

The former are adopted by producers|industrial �rms or agricultural enterprises|

2Cross-country evidence for developing countries suggests that openness to trade facilitates

growth. It has nevertheless proven di�cult to establish a causal link empirically, due to the presence

of many confounding factors in the time series.

3

and then utilized by that producer's workforce, be it one individual or a large group.

Thus, the adoption decision is at the producer level, and the technology is a non-

rival input across that producer's workers. In contrast, ideas are the property of

individuals: an individual can utilize only ideas that he/she has adopted. Ideas are

surely a more fundamental concept: all technological innovations begin with an idea.3

But technologies are|perhaps|easier to measure.

The rest of the paper is organized as follows. Section 2 reviews several detailed

studies of the di�usion of particular technologies across producers in the U.S. Section

3 looks at the evidence on cross-country di�usion, including evidence on productivity

in agriculture. Section 4 looks at two simple models of technology di�usion across

producers that are compatible with much of the evidence. The �rst is suitable for

looking at di�usion within a single country, the second for looking at cross-country

di�usion. Section 5 concludes.

2. EVIDENCE ON TECHNOLOGY DIFFUSION ACROSS U.S.

PRODUCERS

A. Hybrid corn: Griliches (1957)

Griliches's (1957) study of hybrid corn adoption over the period 1933-1956 is, de-

servedly, a classic. The goal of his paper was \to understand a body of data: the

percentage of all corn acreage planted with hybrid seed, by states and by years." As

he notes, there were marked di�erences in the patterns of adoption across geographic

regions. As he also notes, these hybrids were not one-size-�ts-all: they had to be bred

separately for each geographic region. The variety adapted to a neighboring region

was a useful starting point in the hybridization process, but a new variety need to be

3Lucas (2009), Lucas and Moll (2014), and Caicedo, Lucas, Rossi-Hansberg (2019), and Le (2020)

develop models where the di�usion of ideas across individuals is the engine of long run growth.

4

developed for each locality.

Griliches's approach is �rst to parametrize adoption in each particular geographic

region|state, county or district|in a parsimonious way, then to �t the adoption

parameters by region, and �nally to explain the cross-region variation in those para-

meters with a few economic variables.

Adoption is measured as the share of corn acreage planted with hybrid. As shown

in Figure 1 (Griliches's Figure 1) for the state level, adoption is well approximated

by logistic functions, with di�erent start dates, speeds of adoption, and long-run

adoption levels.

Let Pi(t) denote the percentage planted in hybrid in region i at date t: For the

logistic form

Pi(t) =Ki

1 + e�(ai+bi(t�toi )); t � ti;

so adoption is described by the date of origin toi ; the ceiling Ki; and the parameters

(ai; bi). Take the log of the logistic to get

ln fPi(t)= [Ki � Pi(t)]g = ai + bi (t� toi ) ;

so given (Ki; toi ) the parameters (ai; bi) ; can be estimated by OLS.

Griliches's data are from 132 crop reporting districts in 31 states. The date of

origin toi is taken to be the date at which penetration is 10%, and the ceiling Ki for

each region is �t by hand. The logistic curves �t well, with R2's over 0.90 in every

case, over 0.95 in most, and 0.99 in many districts.

The entry dates, the origins toi , are quite variable, ranging from 1935 in some parts

of Iowa and Illinois to 1945 in Oklahoma and 1949 in some parts of Alabama. They

are lowest in the corn belt states, Iowa, Illinois and Indiana, and gradually spread

out to adjacent areas. The corn belt is also where the bi's and Ki's|the slopes and

ceilings|are highest. Additionally, districts where the ceiling is Ki = 100% also have

high and similar slopes, while places with lower ceilings also have slower speeds of

5

adoption.

Griliches then explores the economic determinants of the variation in the parame-

ters toi ; bi; and Ki across regions. Farmers can't adopt if the seed companies in their

region are not o�ering the hybrid, so toi depends on suppliers: the agricultural experi-

ment stations and private seed companies. Griliches's hypothesis is that although the

origin dates depend largely on the actions of suppliers, while the slopes and ceilings

depend on the actions of adopters, the incentives for both sides of the market depend

on pro�tability.

Commercial seed companies in more pro�table regions should have a greater in-

centive to move earlier, including a greater incentive to encourage the agricultural

stations they rely on. Indeed, Griliches �nds that the origins are fairly well described

as functions of two variables that a�ect supply cost: market density|which lowers

marketing costs, and a geographic variable indicating whether entry had already oc-

curred in a contiguous market|which lowers R&D costs. The estimated slope bi;

interpreted as an expected rate of acceptance, is also useful in explaining the origins.

The speed of acceptance bi is well explained by the degree of superiority of hybrids.

Two measure are tried: the increase in yield per acre, from questionnaire data, and

the pre-hybrid yield per acre. Hybrids increased yields by about 20%|at least that

was the belief at the time, so a higher pre-hybrid yield was an indicator for a larger

gain. The long run level of adoption Ki is fairly well explained by the same variables.

In summary, Griliches �nds that hybrid adoption across geographic regions is well

explained by their relative pro�tability across regions. They were most useful in the

Corn Belt, where they were introduced earlier, were adopted more quickly, and had

greater long-run success. Their use branched out gradually to adjacent regions.

6

B. 12 industrial innovations: Mans�eld (1961)

Mans�eld uses a similar methodology to look at the di�usion of twelve major in-

novations in four industries: bituminous coal, iron and steel, brewing, and railroads.

An important new feature of these innovations compared with hybrid corn is that all

except one involved purchases of new heavy equipment. Thus, in each case a major

investment was required to obtain a substantial reduction in costs.

Mans�eld does not have production data, so his Figure 1 shows the percentage

of �rms in the industry that have adopted. Only larger �rms in each industry are

represented, so the share of adopting �rms is used as a proxy for the share of output

produced with the new method.

Di�usion is rather slow|in many cases 20 years or more, and varies widely across

innovations. The time until half of �rms have adopted varies from 0.9 years to 15,

with an average of 7.8.

The technologies in Mans�eld's data arrived at di�erent well-de�ned dates, and

in each case the ultimate adoption rate was 100%, Thus, Mans�eld focuses on the

speed of adoption, the analog of Griliches's parameter bi. He adopts the same over-

all methodology, �rst estimating an adoption speed for each innovation, and then

regressing those adoption speeds on a small set of (economic) explanatory variables.

Mans�eld's data cover di�erent inventions in di�erent industries requiring di�erent

levels of investment in new equipment. To accommodate these three di�erences, he

uses a more complicate regression equation. In particular, he looks at an equation

that relates the change in the share of adopters as a function of pro�tability, the cost

of adoption, and interactions between those variables and the share �rms that have

already adopted.

Let nij denote the number of potential adopters of innovation j in industry i; and

let mij(t) denote the number who have adopted by date t; so mij(t)=nij is the share

7

of �rms that have adopted by t: Measure pro�tability �ij as the ratio of the average

threshold payback period for the industry to the average payback period (as reported

by �rms) for this investment, a measure that is similar to the ratio of the two Internal

Rates of Return. Measure cost Sij as the ratio of the average initial investment

required for adoption relative to average assets of the �rms in that industry.

Let �ij(t) denote the share of non-adopters at t that adopt by t + 1: Mans�eld's

regression equation is

�ij(t) = ai1 + ai2�ij + ai2Sij

+�i1mij(t)

nij+ �i2�ij

mij(t)

nij+ �i3Sij

mij(t)

nij+ :::

� aij + �ijt+ :::

where the second line uses

aij = ai1 + ai2�ij + ai3Sij;

bij = �i1 + �i2�ij + �i3Sij;

and an assumption that the share of adopters increases approximately linearly with

time, mij(t)=nij � t:

Use this approximation to get the logistic,

pij(t) =1

1 + e�(aij+bijt);

where pij(t) is share of �rms that have installed by t; and the \ceiling" is taken to

be 100% adoption, K = 1: Take logs and use OLS, as Griliches did, to estimate aij

and bij for each innovation. The �ts, reported in Table IB, are very good for all the

innovations.

The bij's are then regressed on an industry constant and the pro�tability and cost

8

measures, giving

bij =

8>>>>>><>>>>>>:

�0:29

�0:57

�0:52

�0:59

9>>>>>>=>>>>>>;+0:530�ij � 0:27Sij;

(0:015) (0:014)r = 0:997:

The coe�cients are highly signi�cant and the �t, displayed in Figure 2 (Mans�eld's

Figure 2), is quite good. The coe�cient estimates are somewhat sensitive to the

outlier (tin cans), but keep the right signs even if that point is excluded.

Mans�eld also tries adding some additional regressors:

|presence of durable equipment that will be made obsolete,

|growth rate of industry sales,

|a time trend in the di�usion rate, and

|the phase of the business cycle when the innovation is introduced.

Each has the right sign but none is statistically signi�cant.

C. Tractors: Manuelli and Seshadri (2014)

Manuelli and Seshadri (2014) look at the adoption of tractors. Adoption in this

case was slow, and it was long a puzzle why it was so slow. The authors show that

adoption is well explained by changes in the total costs of the services provided by

tractors and by the alternative source of farm power, horses, including the required

labor input.

As shown in Figure 3 (MS's Figure 1), although tractors were introduced in 1910,

there was very little adoption before 1920. Adoption rose steadily between 1920

and 1960, with the number of horses and mules declining as the number of tractors

increased. Over this period quality-adjusted tractor price fell very substantially, as

shown Figure 4 (MS's Figure 2). The sharpest decline in price came before 1920,

9

however, and did not induce widespread adoption. Wages, on the other hand, which

were about constant until the mid-1930's, then rose sharply until the end of World War

II. Manuelli and Seshadri's hypothesis is that the increase in wages made tractors|

which are labor-saving|more pro�table, and spurred their rapid adoption during

that period.

Their model of the agricultural sector and the demand for inputs is fairly straight-

forward, and it �ts the data well.

1. Markets for agricultural inputs.|

The production function for agricultural output has constant returns to scale, with

tractors k; horses h; a vector of labor inputs n = (nh; nk; ny) and land a as inputs.

It is a nested CES with three layers. The innermost layers combine tractors/horses

with labor inputs to produce the two individual sources of power,

zx =�!xx

��x + (1� !x)n��xx

��1=�x; x = k; h:

Since the weights !x and elasticities 1= (1 + �x) ; x = k; h; can di�er between the two

power sources, changes in the wage rate can have di�erential e�ects on the costs of the

two power sources. The next layer of the CES aggregates the two sources of power,

z =�!zz

��1k + (1� !z) z

��1h

��1=�1; (1)

and the outermost layer is a Cobb-Douglas aggregator of power, labor and land,

y = F (z; ny; a) = Actz�zn�ny a1��z��n : (2)

The inputs (h; n; a) of horses, labor and land are straightforward to measure. The

cost of using a horse in period t is qht+cht; where qht is the rental rate and cht includes

operating costs. Similarly, the cost of using an acre of land is qat + cat; and the wage

rate is wt: The tractor input kt and its cost per period are more complicated.

10

2. Cost of tractor services.|

An important feature of tractors is that later vintages improved in terms of both

durability and attributes. Thus, it is important to distinguish tractors by their vin-

tage �: Suppose attributes can be mapped into an aggregate of `tractor services.'

In addition, assume the market for tractor services is a perfectly competitive rental

market.

For any vintage �; there are machines of various types �; with di�erent prices and

di�erent vectors of attributes. We will ignore that for now, and suppose that only a

single type of new machine is sold at each date.

At any date t; machines of vintage � = t; t � 1; :::; are available for use. For

any vintage �; let �k� be the depreciation rate, and let v(x� ) denote the quantity

of services provided per machine, where x� is a vector of attributes, and the time-

invariant function v maps attributes into an index of services. Let m(�) denote the

number of machines of vintage �: Then the total supply of tractor services at t is4

kt =tX

�=�1(1� �k� )

t�� v(x� )m(�): (3)

Let R�t denote the discount factor between � and t;

R�� = 1;

R�t = R�t�11

1 + rt; t = � + 1; ::::;

where rt is the one-period interest rate at t. Let qkt(�) and pkt(�) denote the rental

rate and price at t for a tractor of vintage � � t. These satisfy the usual no-arbitrage

condition,

pkt(�) = qkt(�) +1� �k�1 + rt+1

pk;t+1(�); all � � t; all t: (4)

4It is easy to allow machines to have a �nite lifetime T:

11

To close the model, assume the price of a new machine of any vintage is related to

its quality by

pk� (�) =1

�v(x� ); all �; (5)

where � can represent any combination of technical change|pushing � above unity,

and variable markup|pushing � below unity.

Let ckt(�) denote the variable cost (fuel, repairs) of operating a tractor of vintage

� at date t � �: Equilibrium in the rental market at t requires the price of tractor

services, call it pskt; be the same for all vintages in use. That is, rental and user costs

satisfy

qkt(�) + ckt(�) = pskt; all � � t; all t; (6)

where pskt; qkt(�) and ckt(�) are measured per unit of tractor services delivered.

Data is available on prices for new and used tractors, pkt(�); for all vintages � � t

and dates t; and on the attribute vectors x� ; depreciation rates �k� ; and number sold

m(�) for all vintages. The interest rates rt and hence the interest factors R�t are also

known. There is no direct information on the rental rates qkt(�) or user costs ckt(�):

To make the model empirically tractable, assume the operating cost depends only

on the date t; so ckt(�) = ckt; all � � t: Then (6) implies

qkt(�) = pskt � ckt

� qkt; all � � t; all t; (7)

so the rental rate qkt at any date is the same for all vintages, and (4) takes the simpler

form

qkt = pkt(�)�1� �k�1 + rt+1

pk;t+1(�); all � � t; all t: (8)

Use (8) for the new vintage at t to �nd that

qkt = pkt(t)

�1� 1� �kt

1 + rt+1

pk;t+1(t)

pkt(t)

�; all t: (9)

12

Next, assume

pk;t+1(t) �1

t+1v (xt) ; all t; (10)

which says the price of a year-old tractor at t + 1 is approximately equal to the

services it provides, adjusted for the new markup t+1: If (5) holds, then (10) is a

good approximation if t does not change too much from year to year. Use (5) and

(10) in (9) to get

qkt = pkt(t)

�1� 1� �kt

1 + rt+1

t t+1

�; all t: (11)

The function v and the values t on the right side of (11) can be estimated and used

to get estimates of the evolution of average tractor quality and the rental rates qkt:

For the estimation, use the fact that at any date t many types � of new tractors

are produced. Let p�kt(t) and x�t denote the price and attribute vector for a particular

new machine �; and assume the function v is log-linear, v(x) = �nj=1 (xj)�j : Use the

data on prices and attributes to estimate the equation

ln p�kt(t) = �dt +nXj=1

�j lnx�jt + ��t ; all �; all t;

where dt is a time dummy. Then use market shares s�kt to calculate the average price

at each date,

pkt =X�

s�ktp�kt; all t: (12)

Use (12) in (11) to get an estimate of the rental rates,

qkt = pkt

�1� 1� �kt

1 + rt+1

t t+1

�; all t; (13)

where t is the estimated time dummy t = edt : The rental rate at t is proportional to

the average quality-adjusted price of a new machine, with a factor of proportionality

that includes the depreciation and interest rates in the usual way, and also includes

the anticipated change in the markup t= t+1: An increase/decrease in the markup

reduces/raises the current rental rate, as owners anticipate a capital gain/loss.

13

The estimate of average quality at t is

v (xt) = tpkt � tX�

s�ktv(x�t ); all t: (14)

3. Cost minimization.|

It is then fairly straightforward to use (1) and (2) to characterize the cost-minimizing

input mix in agriculture. Estimating the model parameters|the elasticities and

shares in the nested CES|from the data on prices and quantities is delicate but

possible.

Figure 5 (MS's Figure 3) displays the resulting estimates for tractor prices pkt and

quality v(xt); as well as the parameter t: Over the period of rapid adoption 1935-

48, tractor quality v(x) rose a little before 1940 and was approximately constant

afterwards. The parameter t rose over most of the period, depressing markups, and

then fell sharply at the end of the war, raising the markup. The result was that the

price was approximately constant over the whole period. Figure 6 (MS's Figure 4)

displays the �t of the model over the entire transition period, which is quite good.

D. Other studies

There are many other papers as well looking at di�usion of particular technologies.

For example, Jovanovic and MacDonald (1994a) look at adoption of diesel locomo-

tives. Invented in 1912 and �rst used in the U.S. in 1925, penetration|as measured

by the share of locomotives that were diesel|was slow for the �rst 20 years, but it

was quite rapid after that, increasing from about 10% in 1945 to almost 90% by 1955

and well over 95% by 1960.

In addition, many papers have studied other aspects of technology adoption. For

example, Gort and Klepper (1982) study the entry and exit of producers in markets

for new products. Using data for 46 products, a mix of producer and consumer goods,

they identify �ve stages, distinguished by entry and exit rates.

14

Jovanovic and MacDonald (1994b) examine industry structure for automobile tires,

and consolidate Gort and Klepper's �ve stages into three: a period with rapid entry,

followed by a \shakeout" period with lots of exit, and then a mature industry with

a moderate number of �rms. The authors develop a simple theoretical model that

produces those three phases, and show that similar industry patterns appear for autos,

airplanes, cell phones.

Grubler (1991) looks at sequences of technologies serving particular functions: the

decline of the horse and rise of the automobile for land transportation; the successive

use of sail, steam and motor propulsion for merchant marine transportation; �ve

methods for steel production; and the di�usion of six new technologies for various

automobile parts.

3. EVIDENCE ON CROSS-COUNTRY DIFFUSION

Detailed data on adoption patterns|arrival and penetration rates|for a broad set

of countries is not available for even a small set of technologies. Thus, data availability

has led researchers to ask a di�erent set of questions. In particular studies of cross-

country di�usion have asked what country characteristics explain faster adoption.

Three approaches have been used.

One approach, used by Comin and Hobijn (2004, 2010), is to look at adoption lags,

in the sense of date of �rst adoption, across a broad set of countries, for a number

of particular technologies. They relate the lags to country characteristics like human

capital and degree of openness. In a related paper, Comin and Mestieri (2017) look

at time trends in both adoption lags and penetration rates.

A second approach, used by Eaton and Kortum (2001), Caselli and Coleman (2001),

and Caselli and Wilson (2004) uses data on imports of capital goods. As noted earlier,

many technologies are `embodied' in capital goods. Thus trade in equipment is,

potentially, an important mechanism by which technologies di�use across countries.

15

And since good bilateral trade data are available at a rather �ne product level,5 data

for capital goods in the relevant categories can be used to ask a number of questions.

A third approach, used by Chen (2018) is to look at TFP growth in agriculture. The

idea here is that long-run TFP growth in that sector is due to technology adoption,

and that many of the technologies relevant for agriculture|tractors, fertilizer, and

so on|are sector-speci�c.

A. Comin and Hobijn (2004, 2010)

Comin and Hobijn (2004) look at the adoption of twenty technologies, in twenty-

three developed countries, over the period 1788-2001. Speci�cally, they look at lags

until �rst adoption. The data are a mix of consumer and producer goods, for some

the data cover a very limited number of countries, and there's much missing data

before 1938.

But four technologies are interesting for our purposes here: personal computers,

measured per capita; industrial robots, measured per unit of GDP; three shipping

technologies|sail, steam, and motor, measured as fraction of tonnage; and four steel

technologies|open hearth, Bessemer, blast oxygen, and electric arc, measured as

fraction of tonnage. In regressions, they �nd that human capital, per capita GDP,

and openness predict earlier adoption.

In Comin and Hobijn (2010) the authors estimate adoption lags for �fteen tech-

nologies, across 166 countries, spanning two centuries. The data here are subject

to the same caveats. They �nd long lags|an average of 45 years, but the lags are

shorter for later technologies.

5For example, see Feenstra, et. al., (2005).

16

B. Comin and Mestieri (2017)

Comin and Mestieri (2017) use an extended version of the data in Comin and

Hobijn (2010) to look further into adoption patterns for twenty-�ve technologies in

139 countries. These technologies are very heterogenous, including producer goods,

consumer goods, and mixed-use goods, and as before much data is missing.6

They �nd that adoption patterns across countries and technologies can be well

approximated in terms of a technology-speci�c shape that is the same across coun-

tries, and two country/time-speci�c parameters that are the same across technologies.

Speci�cally, adoptions patterns for each technology/country pair can be described by

�rst �tting a basic `shape' for adoption in an advanced country where detailed, reli-

able data is available, and then shifting the date of origin and stretching the curve

downward/rightward at later dates for other countries. The lag and penetration pa-

rameters then describe the length of time between the innovation date and the �rst

use in a particular country and how quickly the new technology di�uses after the �rst

adoption.

For the empirical work they aggregate the countries into two groups, Western (ad-

vanced) and non-Western. They estimate the technology-speci�c shapes from U.S.,

U.K., French or German data, and the country/time parameters with a set of decade

� country-group dummies.

Thus, the approach is similar to the one in Griliches (1957) and Mans�eld (1961),

except that the shape is not assumed to be logistic. In addition, the de�nition of

adoption is di�erent. In Griliches (1957) it is the share of acreage and in Mans�eld

(1961) it is the share of (large) �rms. Since Comin and Mestieri do not have data

6The technologies in the three groups are: spindles, ships, railroad freight, aviation freight, trucks,

fertilizer, tractors, harvesters, steel production techniques, synthetic �bers; railroad passengers, avi-

ation passengers, cars, medical procedures (kidney transplants and liver transplants, heart surgery);

and electricity, telegraph, mail, telephones, cellphones, cars, internet.

17

on output at the country/industry/time level, adoption is de�ned as (log) output

produced with the new technology relative to total GDP, or (log) input of the new

technology (e.g. the stock of the new capital good) relative to total GDP. Thus, the

penetration parameter also picks up cross-country di�erences in output composition.

Their procedure produces predicted di�usion paths for each technology � country-

group pair. As a robustness check they divide the non-West group into quartiles,

estimating group-time (decade) �xed e�ects for each group. These do not vary much

across groups.

The authors �nd that over time, adoption lags have gotten shorter but the di�erence

between advanced and less-advanced countries in speed of penetration has gotten

larger. One candidate explanation is that less developed countries have two groups

of �rms operating in parallel: one adopts modern technology|although with a lag,

and the other never adopts.

C. Eaton and Kortum (2001)

Eaton and Kortum (2001) pioneered the study of trade data on equipment imports

and exports. They show that across countries, equipment output as a share of GDP

increases strongly with GDP per capita, and net equipment exports as a share of

GDP increase strongly with equipment production as a share of GDP per capita.

Developing countries, on the other hand, import a large share of their total investment

in new equipment and this equipment comes from a small set of exporters. Thus, new

capital goods are potentially an important mechanism by which technologies di�use

to less developed countries.

For 1985, the year they use for their cross-sectional study, Eaton and Kortum

identify a `big 7' group of countries that accounted for a large share of equipment im-

ports in most countries. The `big 7' countries|U.S., Japan, Germany, U.K., France,

Sweden, and Italy|are also R&D intensive. They show that across a broad set of

18

countries in the rest of the world, imports from the `big 7' accounted for 64-92% of

equipment imports.

Eaton and Kortum also show that the price of equipment relative to consumption

goods is strongly decreasing in GDP per capita. In a variant of a standard growth

model, this fact implies that di�erences in the relative price of internationally traded

equipment is a signi�cant factor in explaining di�erences in cross-country productivity

levels. Here the authors �nd that here it explains about a quarter of those di�erences.

By 2000, China, Taiwan and Korea had overtaken France, Sweden, and Italy to

form a new `big 7' group of exporters (author's calculation). Imports from the new

group form a smaller|but still very substantial|share of total equipment imports in

most countries in 2000. Thus, it seems likely that the mechanism identi�ed by Eaton

and Kortum continues to operate.

D. Caselli and Coleman (2001)

Caselli and Coleman (2001) look at cross-country data on computer adoption over

the period 1970-1990. Most countries produced little or no computer equipment of

their own during this period, especially in the earlier years, so adoption can be proxied

by imports. To check robustness they try a couple of di�erent measures, one of which

involves excluding countries with computer exports.

They �nd that schooling levels have an important e�ect on computer imports.

In the non-exporting sample, a one percentage point increase in the fraction of the

population with schooling above the primary is associated with a �ve percent increase

in computer imports. There is also an important shift over time, presumably because

computers became better and cheaper (Figure 1 in the NBER pre-print).

19

E. Caselli and Wilson (2004)

Caselli and Wilson (2004) develop a simple structural model to look at adoption

of embodied technologies. The model has many sectors, each of which uses sector-

speci�c capital equipment together with homogeneous labor to produce an interme-

diate, where the productivity levels in each sector are country-speci�c. The inter-

mediates are used in a CES function that is common across countries to produce

�nal goods. The e�cient (competitive equilibrium) allocation requires that the rela-

tive capital shares across sectors in each country equal the productivity ratios across

sectors in that country.

As noted before, since most equipment is produced in a small set of countries, for

the rest of the world investment is well proxied by imports. Caselli and Wilson iden-

tify nine important categories of capital equipment|electrical, non-electrical, o�ce,

communication, motor vehicles, etc.|and document large cross-country di�erences

in import shares across categories. For the empirical work, the authors exclude big

exporters and construct stocks of equipment in each sector by combining share data

on capital goods imports and NIPA data on aggregate investment.

The di�erences in shares across categories of equipment can be interpreted as dif-

ferences in adoption rates for di�erent types of technologies. The authors relate these

di�erences in adoption rates to country characteristics. Speci�cally, they conjecture

that the country/sector productivity parameters depend on observable country char-

acteristics. The relative import shares should then also depend on those characteris-

tics. The characteristics they look at are the availability of complementary factors:

educated labor, institutions, composition of GDP, level of �nancial development, and

so on. The coe�cients in the resulting regression equation are the relative impor-

tance of the country characteristics in determining the country-sector productivity

parameters.

20

They �nd that human capital is complementary to computers, electrical equipment,

communication equipment, motor vehicles, and professional goods; and income per

capita is complementary to computers and electrical equipment. Although the results

are sensitive to the set of other regressors that are included, both human capital and

income per capita could be thought of as proxies for wage rates. A time trend is also

signi�cant for some technologies|computers, electrical equipment, communication

equipment, and aircraft, perhaps because they had more signi�cant price declines.

The authors also construct a measure of the R&D intensity for each category of

capital equipment. They �nd that the median country is slower to adopt more R&D-

intensive technologies, but those categories enjoy more rapid increases over time.

Perhaps equipment in those categories is initially less suitable to the median country,

but experiences bigger quality increases and/or price declines over time.

F. Chen (2018) Technology adoption in agriculture

The share of labor employed in agriculture declines sharply as a country develops,

as seen in both cross-country data and time series for developed countries. Figures

7 and 8 (Lucas, 2009, Figures 11 and 13) show the plot for a large cross section of

countries and time series for the U.S., U.K., Japan and India. Thus, improvements in

labor productivity in agriculture and the movement of labor into the non-agricultural

sector are critical for understanding development.

In addition, cross-country di�erences in labor productivity are larger in agriculture

than in nonagriculture (Caselli, 2005; and Restuccia, Yang and Zhu, 2008). Figure 9

(Restuccia, Yang and Zhu's Figure 2) displays labor productivity in the two sectors

(panel a) and productivity in agriculture relative to non-agriculture (panel b).

Motivated by these facts, Chen (2018) studies technology adoption in agriculture,

looking at both cross-country evidence and U.S. time series. Chen starts by con-

structing a cross-country data set on capital intensity by sector, which shows that the

21

patterns seen in labor productivity also hold for capital intensity. Figure 10 (Chen's

Figure 1, panels a and b) displays the cross-sectional capital-output and capital-labor

ratios, measured at international prices, in agriculture and nonagriculture. Chen also

shows that in the U.S., the capital-output ratio in agriculture rose over the period

1920-2000 but showed no trend in nonagriculture. Figure 11 (Chen's Figure 2) shows

the two time series (panel a) as well as plots of the adoption rates for various types

of new capital equipment (panel b).

Chen argues that di�erences between the two sectors in technology adoption ex-

plain these patterns. His model starts from the fact that much technical change in

agriculture is embodied in new equipment: tractors, trucks, combines, balers, and

so on. Chen's two-sector general equilibrium model includes labor allocation, con-

sumption and investment decisions, and features investment-speci�c technical change

(ISTC). Here we will focus on the model's novel feature, which is the treatment of

technology adoption in agriculture.

As in Hansen and Prescott (2002), two technology alternatives are available in

agriculture. Each is Cobb-Douglas, with capital k; land `; and farmer's ability s as

inputs, and di�erent share parameters. Let

y = Axs1��x� xk�x` x ; x = r;m;

denote the production functions for the traditional (r) and modern (m) technologies.

The modern method is assumed to be more capital intensive, �m > �r; and (weakly)

less intensive in both of the other inputs, m � r; and 1� �m � m � 1� �r � r:

There are perfectly competitive markets for the rental of capital equipment and

land. Let Rk and q denote the rental rates, normalize the output price to unity,

and consider the (static) decision problem of a farmer at any date. Farmers are

heterogenous in terms of ability s. It is straightforward to show that the operating

22

pro�t for a farmer with ability s using either method is

�x(s;Rk; q) = maxk;`

�Axs

1��x� xk�x` x �Rkk � q`�

= �0xR��xk q� xs; x = r;m;

where the constant �0x depends on Ax; �x and x: Both pro�t functions are linear in

s; and the elasticity of pro�ts with respect to the user cost of capital is ��x: Thus,

both techniques become more pro�table as Rk falls, but since �m > �r; a change in

Rk has a proportionately larger impact on pro�ts from the modern technology.

If there are no other costs, then for any �xed rental rates (Rk; q) ; either �r > �m

or �x � �m; all s; so either all farmers adopt the modern technology or none do. To

make adoption gradual, Chen introduces a �xed cost of adoption, f for the modern

technology. The choice for a farmer with skill s is then

max f�r(s;Rk; q); �m(s;Rk; q)� fg :

For moderate levels of the �xed cost, farmers with ability s above some threshold s

adopt the modern technology and the rest do not. The left panel of Figure 12 (Chen's

Figure 3), which plots �r(s) and �m(s) for �xed q and two values for Rk; shows the

two possibilities.

Chen models the gradual adoption of modern farming methods through the reduc-

tion in the cost of capital Rk; induced by ISTC. As Rk declines, the slopes �m and

�r both increase, as shown in the right panel of Figure 12, but the former increases

proportionately more. Thus, the threshold s declines as Rk falls, and over time an

expanding set of farmers adopts the modern technology.

It is always the highest ability farmers who adopt the modern technology. Although

ability is not directly observable, land use ` is proportional to ability s for each

technology. In addition, it is straightforward to show that if m = r; and if factor

prices are such that �m > �r; then farmers using the modern technology operate more

23

land than those using the traditional technology. Hence it is larger farms that adopt

modern, capital-intensive methods. Chen shows that this was the case in the U.S.

In Chen's model investment-speci�c technical change (ISTC) also contributes to

TFP growth in the non-agricultural sector, and there are other forces at work as well,

including the relative price of the agricultural output, the price of land and the wage

in the nonagricultural sector. But, consistent with Figure 11, the nonagricultural

sector is assumed to have only one basic technology, so the capital-output ratio in

that sector is roughly constant over time.

4. TWO SIMPLE MODELS OF TECHNOLOGY ADOPTION

In this section we will develop two simple theoretical models of technology adoption.

The �rst is directed toward explaining speeds of di�usion in a single country, the

second toward explaining cross-country di�usion. Both are dynamic models based

on cost reduction. Both draw on Manuelli and Seshadri's (2014) model of tractors

and on Chen's (2018) model of cross-country di�usion of agricultural technologies in

developing countries, as well as on the model in Jovanovic and MacDonald (1994a).

A. Hybrid corn and industrial innovations

Suppose there are two technology levels, indexed by � = 0 (� = 1) for the old

(the new) technology. Assume the interest rate r > 0 is constant over time and that

there is no entry. The state variable is � 2 [0; 1] ; the share of producers (or industry

capacity) that has already adopted the new technology.

For hybrid corn the goal is to explain di�erences in adoption patterns across regions:

adoption was earlier, faster, and more complete in regions where yields were higher

and acreage in corn was larger. For this case let i = 1; ::; I, denote regions. For the

industrial technologies in Mans�eld's study, the date of �rst adoption varied exoge-

24

nously across innovations and long-run penetration was complete in all cases. Hence

the only goal is to explain di�erences in the speed of adoption across innovations:

controlling for industry, adoption speed was positively related to incremental net rev-

enue per unit of capacity from adoption, and negatively to the required investment.

For this case let i = 1; :::; I; denote the various innovations.

1. Model components.|

For both hybrid corn and the industrial examples, the key model elements are the

size distribution of production units, the pro�tability per unit of capacity of using the

old and new technologies, and the �xed (sunk) cost of adopting the new technology.

For corn, farms vary in size|acreage|within each region, and the size distribution

varies across regions. Let Fi(z) denote the CDF for acreage in region i; for i = 1; :::; I:

The operating pro�t per unit of capacity is the pro�t per acre from growing the old

variety and the hybrid. Let �i1 > �i0 > 0 denote operating pro�ts|revenue net of

variable costs|per acre in region i with the hybrid and the old variety. Corn from

all regions is sold on a common domestic market, so the output price p is the same

across regions. Suppose in addition that p is constant over time. Both yield per acre,

call it yi; and the seed and other input costs, call them i; vary across regions.

No capital investment is required to grow hybrid corn, but the seeds and other

inputs are more expensive, and the hybrid raises yield per acre. Suppose the hybrid

increases yield by a roughly constant percentage gy across regions, and increases costs

by a common constant g : Then pro�ts per acre for the old variety and the hybrid

are

�i0 = pyi � i;

�i1 = (1 + gy) pyi � ( i + g ) ; i = 1; :::; I;

25

and the incremental pro�t per acre

�i � �i1 � �i0

= gypyi � g ; i = 1; :::; I;

is higher in regions i where yield yi is higher.

Absent other factors, all producers in region i would adopt immediately if �i > 0;

and otherwise none would ever adopt. To explain gradual di�usion, suppose there

is a one-time �xed (sunk) cost of adoption, interpreted as the cost of learning about

the growing method for the new seed. Suppose further that the �xed cost falls with

the share of other farms or acreage in the region that has already adopted. The

interpretation is that farmers learn about the new growing requirements from their

neighbors within the region. Let � 2 [0; 1] denote this share, and assume the adoption

cost function c0(�) is the same across regions. Otherwise all di�erence across regions

could be explained as di�erence in adoption costs.

For the industrial examples, measure size by capacity|output or sales (revenue)

per year|or by employment, and let Fi(z) denote the CDF for the size distribution

of producers in industry i. In this case price pi depends on the industry i; but \yield"

is simply unity.

Suppose as before that the new technology increases output per unit of capacity

by a factor gip and changes variable cost by gi , where those changes obviously vary

by industry. Then pro�ts per unit of capacity in industry i are

�i0 = pi � i;

�i1 = (1 + giy) pi � ( i + gi ) ;

and the incremental pro�t of adoption is

�i = �i1 � �i0

= giypi � gi ; all i:

26

In this context gi can be negative, if the innovation reduces material, energy, or

other input costs.

The industrial innovations required substantial investments in new equipment, so

the adoption cost depends on the �rm's size as well as the share of earlier adopters.

As before, later adopters can learn from those who adopt earlier, here by direct

communication between �rm managers or by poaching their workers. Assume the

cost of adoption for innovation i in a �rm with capacity z; as a function of the

penetration rate, is ci0(�)+ ci1z:7 The �rst component represents the cost of learning

the new method, and the second represents the cost of the new equipment.

In summary, in each case the model inputs are the size distributions Fi and para-

meters f�i0; �i1; ci1g and function ci0(�) describing pro�ts and costs. For hybrid corn

c0(�) does not vary with i and ci1 = 0, all i:

There are no interactions across regions/industries in either case. For the industrial

innovations this fact is obvious. For hybrid corn it is a consequence of the assumption

that the price p does not vary with production. Thus, competitive equilibrium in

each region/industry i involves only the evolution of the penetration rate �i and the

resulting evolution of the adoption cost ci0(�i).

2. Competitive equilibrium.|

For notational simplicity we drop the subscript i in this subsection. Let r denote

the interest rate. Fix F; with support Z = [zmin; zmax], the parameters (�0; �1; c1; r) ;

and the function c0(�). We will maintain the following restrictions throughout.

Assumption 1: a. F has a continuous, strictly positive density on Z = [zmin; zmax];

b. �0 > 0; �1 > 0; c1 � 0; r > 0;

c. c0(�) is continuous and strictly decreasing in �:

7Industry price pi; and the price of equipment ci1 might also decline as adoption proceeds. In

this case an additional assumption is required, restricting the relative rates of decline.

27

Let G � 1 � F denote the right tail CDF for F: Assume that the cost of adoption

c0(�) depends on the penetration rate in the previous period.

Informally, in terms of sequences a competitive equilibrium is de�ned by a nonde-

creasing sequence of penetration rates ��1 = 0 and f�ng1n=0 ; and associated thresholds

z0 = G�1(0) = zmax and fzn = G�1(�n�1)g1n=0 ; with the following property: for each

n = 1; 2; :::; if the penetration rate is �n�1 at the end of the previous period and is

expected to be f�n+ig1i=0 in the current and subsequent periods, then it is optimal for

�rms with z � zn to adopt in the current period if they have not already done so and

for all others to wait.

Since the sequence f�ng is nondecreasing, it reaches or approaches an upper bound.

Let � � limn!1 �n � 1 denote this bound. If the penetration rate at the beginning

of the period is �; then either all �rms have adopted already (� = 1), or no �rms with

z < G�1(�) �nd it pro�table to adopt.

Alternatively, since adoption involves an intertemporal trade-o�, we can take a

recursive approach. Then the individual state variable for each producer is his size z;

and the aggregate state variable is the penetration rate � at the end of the previous

period. Let V�(z; �); � = 0; 1; denote the value of a producer of size z when the state

is �; and the producer has not (� = 0) or has (� = 1) adopted the innovation.

We then have the following de�nition.

Definition: Given (F; �0; �1; c0; c1; r) ; a competitive equilibrium is pair of value

functions V�(z; �); � = 0; 1; all z; �; and a nondecreasing function �(�) with �(�) � �;

all �; describing the law of motion for �; such that:

i. for all z; �;

V1(z; �) = �1z +1

1 + rV1(z;�(�)); (15)

ii. for all z; �;

V0(z; �) = max

�V1(z; �)� [c0(�) + c1x] ; �0z +

1

1 + rV0(z;�(�))

�; (16)

28

iii. for all �; the �rst option (the second option) in (16) is optimal for z � �(�)

(for z � �(�)).

Producers who have already adopted make no more decisions, and (i) is the Bellman

equation for their value function. Producers who have not yet adopted must decide

whether to adopt in the current period or wait, and (ii) is the Bellman equation for

their decision problem. Condition (iii) states that � is the equilibrium law of motion

for the state variable �:

Notice that producers do not necessarily adopt on the �rst date when adopting

this period dominates never adopting. The continuation value V0(z;�(�)) includes

the option of adopting later, and since the �xed cost falls over time, that option

is valuable. Adopting later delays the arrival of the gain, but it also reduces the

adoption cost.

Next we will show that a competitive equilibrium exists and it is unique.

3. Long-run penetration.|

First consider the long-run penetration rate, call it �:When penetration has reached

this level, all �rms of size z � z � G�1(�) have adopted, while the rest have not

adopted in the past and choose not to adopt this period. To establish the existence

and uniqueness of a value � > 0 with these properties, a little more structure is

required.

Let � � � � �c1; denote the net one-period gain from adoption, where � �

r= (1 + r) annuitizes the equipment cost.

Firms gain from adoption if and only � > 0: Assume this holds. Assume in addition

that the largest producers prefer to adopt, even if no others do so.

Assumption 2: a. � = �� �c1 > 0:

b. ��1�zmax > c0(0):

To characterize �; �rst note that any penetration rate e� where �rms of size z �29

G�1(e�) are willing to adopt if they have not done so already and the rest are not,satis�es

��1�z T c0(e�); as z T G�1(e�); all z 2 Z:

Under Assumptions 1 and 2 this condition holds if and only if

�G�1(e�) � �c0(e�); w/ eq. if e� < 1: (17)

Note that (17) holds with equality if long-run penetration is less than complete. We

then have the following result.

Proposition 1: If Assumptions 1 and 2 hold, there exists at least one value e� > 0satisfying (17).

Proof: If �G�1(�) > �c0(�); all � 2 [0; 1] ; then e� = 1 satis�es (17). Otherwise,by Assumption 2b the (strict) inequality holds at � = 0; and the reverse (weak)

inequality holds for some � 2 (0; 1]: Since both G�1 and c0 are continuous in �; it

follows that �G�1(e�) = �c0(e�) for some e� 2 (0; 1]: �

4. Dynamics of adoption.|

Next consider the dynamics of adoption. A �rm that has not yet adopted can adopt

this period or next, or can wait at least two periods. Use (15) and (16) to �nd that

under Assumption 2, adopting this period is preferred to waiting one period if and

only if

�z � c0(�)�1

1 + rc0(�(�)): (18)

The one-period gain from adoption must outweigh the reduction in the �xed cost

from waiting one period.

Since � > 0; if (18) holds for z; it also holds for all z > z: We then have the

following result.

Proposition 2: Under Assumptions 1 and 2 there exists a unique competitive

30

equilibrium adoption function �. The long-run penetration rate � is the minimum

value satisfying (17), and there are two cases for �.

Case a: If � = 1 and �zmin > �c0(1); long-run penetration is complete and occurs

in a �nite number of periods. In this case ��1(1) = [�; 1] ; where � satis�es

�zmin +1

1 + rc0(�) � c0(1); w/ eq. if � > 0; (19)

and ��1; for � 0 2 [�1; �) is the single-valued function de�ned by

�G�1(� 0) +1

1 + rc0(�

0) = c0(��1(� 0)); (20)

where �1 > 0 satis�es

�G�1(�1) +1

1 + rc0(�1) = c0(0): (21)

Case b: If �G�1(�) = �c0(�); the penetration rate approaches � asymptotically,

and if � < 1, long-run penetration is less than complete. In this case ��1 is the

single-valued function de�ned by (20) for � 0 2 [�1; �]; where v1 is de�ned in (21).

Proof: The proof is constructive. Proposition 1 establishes that (17) has at least

one solution. Since all solutions lie on the interval [0; 1] ; the set of solutions has a

minimum. We will show below that this value is �:

Case a: To construct ��1(1); suppose the penetration rate is expected to increase

to � = 1 in the current period. Adopters must prefer (at least weakly) to adopt when

the �xed cost is currently c0(�); � 2 ��1(1) and is expected to fall to c0(1):

Assumption 1c implies immediately that ��1(1) is a closed interval: it has the form

[�; 1] ; and since G�1(1) = zmin is the critical �rm, it follows from (18) that � satis�es

(19). If (19) holds at � = 0; the construction is complete.

Otherwise, � > 0 and the rest of the construction is similar, except that ��1 is

single-valued. From (18), if the penetration rate is expected to increase to � 0 in the

current period, ��1(� 0) satis�es (20). Since G�1 and c0 are continuous and strictly

31

decreasing, so is the left side of (20). Hence ��1:[�1; 1) ! [0; �) de�ned by (20) is

continuous and strictly increasing, and �1 � �(0) satis�es (21).

Moreover, since � 0 < �; and � is the minimum value satisfying (17), it follows that

�G�1(� 0) < �c0(�0) =

�1� 1

1 + r

�c0(�

0); all � 0 < �:

Since c0 is a decreasing function, it then follows from (20) that ��1(� 0) < � 0; all

� 0 < �:

Case b: If �G�1(�) = �c0(�); the argument is the same except that J is de�ned

on [�1; �] ; and ��1: [�1; �]! [0; �] is everywhere single-valued. The penetration rate

approaches � asymptotically, and long-run penetration is less than complete if � < 1:

�

Producers with z < G�1(�) never adopt the new technology. Nor do they exit:

since there is no �xed cost of operating, they simply continue producing with the old

technology. If there were a �xed cost, smaller �rms might exit if �0(�) falls, or they

might be bought out or consolidated into larger units. These considerations will not

be explored here, although the present model could be extended to include them.

5. Empirical predictions: e�ects of parameter changes.|

The substantive conclusions in the studies by Griliches and Mans�eld involved the

economic factors leading to faster or slower penetration, and more or less complete

penetration, across regions and industries. Griliches found that both the speed of

adoption and the long-run level of use|the slope and ceiling of the logistic function,

were increasing in the pro�tability of hybrids.8 Mans�eld, focusing on the slopes only,

found that they were well explained by pro�tability and the initial investment cost.

8Griliches also found that the date of �rst adoption was earlier in regions where the hybrid was

more pro�table. The date of �rst adoption depended on the behavior of suppliers, the USDA and

the seed companies. The model here does not include suppliers, and has nothing to say about dates

of �rst adoption.

32

To ask if the model produces these patterns, at least qualitatively, we must look

at the e�ects of parameter changes. Pro�tability net of the equipment investment

cost is re ected in the parameter � = �� c1: For the hybrid corn example it is also

re ected in the distribution function F; since farms with higher acreage in corn have

greater total bene�ts from adoption and a correspondingly greater incentive to incur

the �xed cost. Hence we are interested in the e�ects of an increase in � or a rightward

shift in F|an upward shift in G�1:

The long-run penetration rate is captured by the equilibrium value for �: The speed

of adoption is re ected in the function �(�); with an upward shift implying faster

and more complete adoption.

For a uniform distribution the equilibrium can easily be calculated (see the Appen-

dix). Let [m;m+ d] denote the support of F: Then an increase in m or d implies an

upward shift. First consider the long-run penetration rate �: There are two possibil-

ities, depending on the parameter values,

� =d�� (�a0 �m�)

d�� �b0� 1 if �a0 �m� � �b0;

� = 1 if �a0 �m� < �b0:

Thus, increases inm and � expand the region where long-run penetration is complete,

and � is strictly increasing in m; d and � in the region where � < 1.

Next consider the dynamics. In the region of parameter space where � = 1; all

producers adopt in the �rst period.9 This outcome is the consequence of the uniform

distribution. In the region where � < 1; the equilibrium adoption function is linear

�(�) = �1 +b0

d� + b0= (1 + r)�;

where

�1 =(m+ d) �� �a0d� + b0= (1 + r)

2 (0; 1) :

9The only exception is the knife-edge case �a0 � �m = �b0, where long-run penetration is

complete, � = 1; but adoption is gradual.

33

The intercept �1; �rst-period adoption, is strictly increasing in m; d; and �; and �

crosses the 45o line at �: Thus, an increase in m; d or � shifts � upward. For this

family, higher pro�tability implies that adoption is faster and more complete.

Figure 13a displays � for a baseline set of parameters and for increases, one at a

time, in m; d and �: Figure 13b illustrates time paths for adoption for the same set

of examples.

B. Cross-country di�usion

Most new technologies are developed in high-income countries, where wage rates

are high. In addition, capital equipment has enjoyed a long and very signi�cant price

decline relative to consumer goods in these countries. Hence many new technologies

are designed to be labor-saving and capital-using. Since in many cases all countries

purchase equipment from the same advanced-country supplier(s), all enjoy the price

declines. Thus, we can expect di�erences in wage levels and wage growth rates across

countries, as well as the declining price of equipment, to be important for understand-

ing the patterns of international technology adoption. Those are the key features of

the model here.

The cross-country evidence evidence is clearest for personal (and other) computers,

showing that higher levels of schooling and higher per capita GDP lead to earlier

adoption. Caselli andWilson (2004) also �nd that higher human capital leads to faster

adoption of electrical and communications equipment, motor vehicle, and professional

goods.

1. The model.|

For simplicity the model here looks at the case where the old technology uses only

labor, and the new one uses both labor and capital.

Index countries by j = 1; :::; J; and assume the new technology is introduced at

34

date t = 0: In addition, assume that the cost of equipment at any date t; call it q(t);

is the same across countries. Suppose the interest rate r and the cost of maintenance

to o�set depreciation � are also the same across countries, and for simplicity assume

they are constant over time. Then the user cost of equipment purchased at t, call

it R(t) = (r + �) q(t); is similar across countries and declines over time if q falls.

But wages wj(t) and their growth rates vary across countries. Thus, the gains from

adoption at any date are smaller, and perhaps nonexistent, in countries with lower

wages levels and slower growth rates.

Suppose the production functions for the two technologies are

y0j = A1��j `�0 ; y1j = (BAj)1�� �k�1 `1��1

��:

For simplicity, returns to scale � are the same for the two technologies, and the shifter

B > 1 is the same across countries. The constant Aj, which encompasses both overall

productivity and a Lucas span-of-control parameter, may vary across countries.

Let p be the output price. For simplicity assume it is the same across countries and

constant over time. It is straightforward to show that pro�ts for the two technologies

are

�0(wj;Aj) = Aj�0w��j ;

�1(wj; R;Aj) = BAj�0C��w1��j R�

���;

where

� � �= (1� �) > 0;

�0 � (p�)1=(1��) > 0;

C� ���� (1� �)1��

��> 0;

are constants, wj is the current wage, and R depends on the price q when the equip-

35

ment was purchased. The gain from adoption is then

��(wj; R;Aj) = Aj�0w��j

�BC�

�wjR

���� 1�:

The term in brackets is increasing in wj; decreasing in R; and can have either sign.

It is positive if and only if the ratio wj=R is su�ciently high.

If the wage wj grows over time or the equipment price q falls or both, adoption may

become worthwhile, even if it is not pro�table when the innovation is �rst introduced.

The optimal adoption date � �j maximizes the PDV of current and future net gains.

2. Optimal adoption.|

Suppose an in�nite horizon. If wj grows at the constant rate gj; and q falls at the

rate : Then � �j solves

max��0

Z 1

�

e�rs (wj0egjs)��

"BC�

�wj0e

gjs

R0e� �

���� 1#ds; (22)

and the condition for an optimum is

� �j �1

�� ( + gj)

��� ln

R0wj0

� lnBC� + lnr + (1� �) �gj

r + (1� �) �gj � ��

�; (23)

w/ eq. if � �j > 0:

Thus, � �j is decreasing wj0, with ��j = 0 if and only if wj0 is su�ciently large. In

addition, � �j is decreasing in gj:

Hence countries with higher wage levels and faster wage growth adopt sooner, as

in available data.

5. CONCLUSION

Di�usion rates vary widely across technologies and countries. Although some rea-

sons for faster or slower adoption are idiosyncratic to a particular technology or

location, as social scientists we are interested in the common factors, the factors that

36

apply broadly. Here we have focused on the dynamics of �xed adoption costs and

di�erences in wage rates.

These cost considerations are surely important in explaining adoption patterns, but

are many other factors as well. There is substantial evidence that new technologies

are complementary with higher human capital, a better educated workforce.10 If

educational attainment in a country is low, that fact can act as a barrier to technology

adoption. Scarcity of complementary factors can also act this way.

There may be other barriers as well, such as vested interests that will take capital

losses if old equipment must be scrapped, as in Parent and Prescott, (1999). These

barriers may take the form of regulations or tari�s that are imposed to protect in-

e�cient domestic producers using older technologies. For technologies that require

substantial investment in new equipment, quality improvements and price declines

over time can lead to additional strategic reasons for delay.

Technology di�usion has many aspects. The cost issues studied here are surely not

the only considerations, but they play an important role in many cases.

10See Nelson and Phelps (1966) for an early exposition of this idea, and Acemoglu (2002) for a

more recent development.

37

REFERENCES

[1] Acemoglu, Daron. 2002. Directed technical change, Review of Economic Studies 69(4):

781-809.

[2] Caicedo, Santiago, Robert E. Lucas, Jr. and Esteban Rossi-Hansberg. 2019. Learn-

ing, career paths, and the distribution of wages, American Economic Jour-

nal:Macroeconomics 11(1): 49-88.

[3] Caselli, Francesco. 2005. Accounting for income di�erences across countries, Handbook

of Economic Growth, 1: 679-741.

[4] Caselli, Francesco and Wilbur John Coleman II. 2001. Cross-country technology di�u-

sion: the case of computers, American Economic Review, Papers and Proceedings

91(2): 328-335.

[5] Caselli, Francesco and Daniel J. Wilson. 2004. Importing technology, Journal of Mon-

etary Economics 51: 1-32.

[6] Chen, Chaoran. 2018. Technology adoption, capital deepening, and international pro-

ductivity di�erences, working paper, National University of Singapore.

[7] Comin, Diego and Bart Hobijn. 2004. Cross-country technology adoption: making

theories face the facts, Journal of Monetary Economics 51: 39-83.

[8] Comin, Diego and Bart Hobijn. 2010. An exploration of technology di�usion, Amer-

ican Economic Review, 100(6): 2031-59.

[9] Comin, Diego and Marti Mestieri. 2018. If technology has arrived everywhere, why has

income diverged? American Economic Journal:Macroeconomics 10(3): 137-178.

[10] Eaton, Jonathan and Samuel Kortum. 2001. Trade in capital goods, European Eco-

nomic Review 45:1195-1235.

38

[11] Feestra, Robert C., et. al. 2005. World trade ows: 1962-2000. NBER Working paper

11040.

[12] Gort, Michael and Steven Klepper. 1982. Time paths in the di�usion of product

innovations. Economic Journal 92(367): 630-653.

[13] Griliches, Zvi. 1957. Hybrid corn: an exploration in the economics of technical change,

Econometrica 25(4): 501-522.

[14] Grubler, Arnulf. 1991. Di�usion: long-term patterns and discontinuities. Technological

forecasting and social change 39: 159-180.

[15] Hansen, Gary D. and Edward C. Prescott. 2002. Malthus to Solow American Eco-

nomic Review 92(4): 1205-1217.

[16] Jovanovic, Boyan and Glenn MacDonald. 1994a. Competitive di�usion, Journal of

Political Economy 102(1): 24-52.

[17] Jovanovic, Boyan and Glenn MacDonald. 1994b. The life-cycle of a competitive in-

dustry, Journal of Political Economy 102(2): 322-347.

[18] Le, Tung. 2020. Immigrants, language barrier, and assimilation. Essays on Economics

of Immigration, Ph.D. thesis, University of Chicago.

[19] Lucas, Robert E. Jr. 2009a. Ideas and growth. Economic Journal 76:1-19.

[20] Lucas, Robert E. Jr. 2009b. Trade and the di�usion of the Industrial Revolution,

American Economic Journal: Macroeconomics 1(1): 1-25.

[21] Lucas, Robert E. Jr. and Benjamin Moll. 2014. Knowledge growth and the allocation

of time, Journal of Political Economy 122(1): 1-51.

[22] Mans�eld, Edwin. 1961. Technical change and the rate of imitation, Econometrica

29(4): 741-766.

39

[23] Manuelli, Rodolfo E. and Ananth Seshadri. 2014. Frictionless technology di�usion:

the case of tractors, American Economic Review 104(4): 1368-91.

[24] Nelson, Richard R. and Edmund S. Phelps. 1966. Investment in humans, technological

di�usion, and economic growth, American Economic Review 56(1/2): 69-75.

[25] Parente, Stephen L. and Edward C. Prescott. 1999. Monopoly rights: a barrier to

riches, , American Economic Review 89(5): 1216-1233.

[26] Restuccia, Diego, Dennis Tao Yang, and Xiaodong Zhu. 2008. Agriculture and ag-

gregate productivity: a quantitative cross-country analysis, Journal of Monetary

Economics 55: 234-50.

40

APPENDIX

A. Example in Figure 13

Assume z � U[m;m+ d], so G�1(�) = m + (1� �) d: In addition assume the

�xed-cost function has the linear form c0(�) = a0 � b0�:

Recall that � is the minimal value satisfying (17). Ass. 2b requires (m+ d) � > �a0:

Assume in addition that �d > �b0: Then � satis�es

� � � (m+ d)� �a0�d� �b0

=�d� (�a0 � �m)

�d� �b0; w/ eq. if � < 1;

so

�a0 � �m � �b0 =) � = 1;

�a0 � �m > �b0 =) � 2 (0; 1) :

Higher values for � and m expand the region where � = 1; and in the region where

� < 1; clearly � is strictly increasing in m: In that region,

d�

d�_ (m+ d) (�d� �b0)� d [� (m+ d)� �a0]

= d� (a0 � b0)�m�b0

> m (d�� �b0) > 0;

so � is also increasing in �; and a similar calculation shows it is increasing in d:

Next consider the dynamics of penetration. In the region where � = 1; use (19) to

�nd that � � 0 satis�es

m� +1

1 + r(a0 � b0�) � a0; w/ eq. if � > 0;

or

� � (1 + r) 1b0(m�� �a0) ; w/ eq. if � > 0: (24)

41

If �a0 �m� = 0; (24) holds with equality at � = 0: In the rest of the region, where

�a0 �m� 2 (0; �b0]; the right side of (24) is negative, so the solution is still � = 0:

Evidently a linear �xed-cost function, c0(�) = a0 � b0�, together with a uniform

distribution for z; implies that if penetration is complete, it occurs in the �rst period.

In the region where � < 1; (20) and (21) imply that �1 > 0 satis�es

� [m+ (1� �1) d] +1

1 + r[a0 � b0�1] = a0;

or

�1 =(1 + r) � (m+ d)� ra0

(1 + r) �d+ b0

=� (m+ d)� �a0�d+ b0= (1 + r)

:

Clearly �1 is increasing in m: Since

d�1d�

/ (m+ d)

�d� +

b01 + r

�� d [(m+ d) �� �a0]

= (m+ d)b01 + r

+ d�a0 > 0;

it is also increasing in �; and a similar calculation shows it is increasing in d: Hence

�rst-period adoption is strictly increasing in all three parameters, m; d;�:

For the subsequent dynamics, use (20) to �nd that ��1 satis�es

� [m+ (1� � 0) d] +1

1 + r(a0 � b0�

0) = a0 � b0��1(� 0):

Inverting gives the equilibrium law of motion for the penetration rate,��d+

b01 + r

��(�) = � (m+ d)� �a0 + b0�;

or

�(�) = �1 +b0

�d+ b0= (1 + r)�:

Hence � is linear in �; it has an intercept at �1 and a slope less than unity, and for

each �xed set of parameters it crosses the 45o line at �: The slope is decreasing in �

and d; and does not depend on m:

42

Figure 1

Figure 2

1910 1920 1930 1940 1950 1960

0

1,000

2,000

3,000

4,000

5,000

Tra

ctor

s

0

5,000

10,000

15,000

20,000

25,000

Hor

ses

and

mul

es

Horses and mules

Tractors

Figure 1. Horses, Mules, and Tractors in Farms: 1910–1960

Figure 3

1910 1920 1930 1940 1950 19600

20

40

60

80

100

120

Rea

l pric

e of

hor

ses

and

trac

tors

Horse

Tractor - quality adjusted

Tractor

0.2

0.7

1.2

1.7

2.2

2.7

Rea

l wag

e ra

te

Wage

Figure 2. Real Prices for Tractors, Horses, and Labor: 1910–1960

Figure 4

0

50

100

150

200

250

1920 1925 1930 1935 1940 1945 1950 1955

γ

Price

v(x)

Figure 3. Tractor Prices, Quality, and Productivity: 1920–1955: Estimation

Figure 5

i h li i l k i h l f b i i l i (

7.5

8

8.5

9

9.5

10

10.5

0

1

2

3

4

5

6

7

8

9

1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960

Tractors - model

Tractors - data

Horses- data

Horses- model

Figure 4. Transitional Dynamics: 1910–1960: Tractors and Horses

Figure 6

12 AMERICAN ECONOMIC JOURNAL: MACROECONOMICS JANUARY 2009

Figure 11. Agricultural Employment Shares, 1980

Figure 7

14 AMERICAN ECONOMIC JOURNAL: MACROECONOMICS JANUARY 2009

Figure 13. Employment Shares in Agriculture

Figure 8

ARTICLE IN PRESS

1 2 3 4 5 6 7 8 9 100

1

2

3

x 104 Panel A: GDP per Worker in each Sector

Agriculture

Non-Agriculture

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

Panel B: GDP per Worker in Agriculture relative to Non-Agriculture

Deciles of Aggregate GDP per Worker

Fig. 2. Sectoral labor productivity across countries—1985. Countries are ranked according to aggregate GDP per worker from PWT5.6

where decile 10 groups the richest countries. Each decile contains eight countries (10% of countries in our sample) except decile 5, which

contains 13 countries.

D. Restuccia et al. / Journal of Monetary Economics 55 (2008) 234–250236

Figure 9

Figure 1: The Capital Intensity across Countries

(a) Real Capital-Output Ratio (b) Capital-Labor Ratio

Figure 10

Figure 2: U.S. Historical Facts

1920 1940 1960 1980 2000

Year

0

1

2

3

4

5

Cap

ital-O

utpu

t Rat

io

AgricultureNon-Agriculture

(a) Capital-Output Ratio

1920 1940 1960 1980 2000

Year

0

0.2

0.4

0.6

0.8

1

Tec

hnol

ogy

Ado

ptio

n R

ate

Grain and bean combinesMower conditionersPickup balersTrucksTractors

(b) Technology Adoption

Note: Figure (a) shows the capital-output ratio in the U.S. measured using current prices and the data are

from the U.S. Bureau of Economic Analysis (BEA). Figure (b) shows the percentage of agricultural output

produced by farms with modern machinery, calculated using data from the U.S. census of agriculture.

percentage of agricultural output produced by farms with modern machinery, such as trucks,

tractors, and combines, in the U.S. starting from 1920. Machinery usage increases rapidly

between 1940 and 1980, which is also the period that agricultural capital intensity increases

relative to the non-agricultural sector. Note that although capital-output ratio increases in

agriculture, the aggregate capital-output ratio is still relatively stable since agriculture is a

small sector contributing to less than 10% of GDP in the U.S. when mechanization starts.

In this paper, I use the historical pattern of mechanization of the U.S. as a benchmark

to study the lack of mechanization in poor countries. To determine whether agricultural

production is under-mechanized in poor countries and why so, it is improper to directly

compare the level of mechanization of poor countries today with that of the U.S. decades

ago when its labor productivity was as low, since they differ along several dimensions. First,

improved modern technologies are already available to poor countries today but not to the

U.S. decades ago.14 Second, various frictions in poor countries, such as lacking land tenure,

prevent mechanization, while these frictions did not exist in the U.S. Finally, poor countries

14For example, Manuelli and Seshadri (2014) argue that when the U.S. farmers started to adopt tractors,they were of low quality, which explains the slow adoption of tractors in the U.S. On the contrary, farmers inthe poor countries today may potentially have access to tractors that are better than what their Americancounterparts used decades ago.

11

Figure 11

Figure 3: Technology Choice

s0

−pkf

πm

πm − pkf

πr

π

s0

−pkf

πm

πm − pkf

πr

π

s

Figure 12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

d = 1.2

m = 1.2

= 1.2

baseline

45o

(

)Figure 13a: equilibrium adoption function

0 2 4 6 8 10 12 14 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

d = 1.2

m = 1.2

= 1.2

baseline

= 1, m = 1, d = 1

a0 = 1.5, b

0 = 1, r = 0.05

time

pene

trat

ion

rate

Figure 13b: Dynamics of penetration